Problem 16 - Introductory Algebra for College Students

Question

Evaluate each function at the given values. \(g(x)=x^{2}+7 x\) a. \(g(2)\) b. \(g(-2)\) c. \(g(0)\)

Step-by-Step Solution

Answer

The results of the calculations are \(g(2) = 18\), \(g(-2) = -10\), and \(g(0) = 0\).

Step 1: Substituting x = 2

Substitute x = 2 into the function \(g(x)\). After substitution, the function becomes \(g(2) = 2^2 + 7*2 = 4 + 14 = 18.\)

Step 2: Substituting x = -2

Substitute x = -2 into the function \(g(x)\). After substitution, the function becomes \(g(-2) = (-2)^2 + 7*(-2) = 4 - 14 = -10.\)

Step 3: Substituting x = 0

Substitute x = 0 into the function \(g(x)\). After substitution, the function becomes \(g(0) = 0^2 + 7*0 = 0.\)

Key Concepts

Function Evaluation

When it comes to understanding algebra, one of the key skills is function evaluation. Essentially, this is the process by which we find the output of a function given a specific input. To evaluate a function like g(x) = x^2 + 7x, you need to replace the variable x with the given number and perform the arithmetic operations as shown in the exercise.

In the given exercise, when evaluating at g(2), we substituted x with 2 and followed the order of operations: first we squared the 2, obtaining 4, and then added 7 times 2 to get 18. The same method applies for g(-2) and g(0), with each substitution yielding -10 and 0, respectively. This direct substitution is the foundation of function evaluation and a fundamental algebraic skill.

Quadratic Functions

A quadratic function is a second-degree polynomial function of the form f(x) = ax^2 + bx + c, where a, b, and c are constants, and a ≠ 0. The graph of a quadratic function is a parabola, which opens upward if a > 0 and downward if a < 0.

The quadratic function in our example, g(x) = x^2 + 7x, is simpler since it is missing the constant term c. Understanding the shape and direction of the graph can aid in anticipating the function's behavior and in understanding the significance of the function's evaluation at different points. For example, a positive or negative result from function evaluation tells us whether the graph of the function lies above or below the x-axis at that point.

Substitution Method

The substitution method is a technique used not just for evaluating functions, but also for solving equations and systems of equations. It involves replacing variables with numbers or other expressions.

In evaluating the function g(x) from the exercise, we employed the substitution method by replacing x with the given values. This method allows us to see the effect of different inputs on the output of the function. It is essential in algebra to master the substitution method, since it also enables us to understand how the parts of a function work together to produce a result, leading to a deeper understanding of function behavior and, more broadly, the concept of a function itself.